Solve simple trigonometric equations in a given interval, including…

The idea

Solving trigonometric equations requires finding all angles in a specified interval that satisfy an equation. For simple equations like sin θ = k, use the inverse function to find the principal value, then use symmetry and periodicity of the trigonometric functions to find all solutions in the interval. For quadratic equations in sin θ, cos θ or tan θ, rearrange to standard form, factorise or use the quadratic formula, then solve each resulting simple equation. Equations involving multiples of the unknown angle, such as sin 2θ = k or cos(3θ + 30°) = k, are solved by first finding the general solution for the multiple angle, then dividing or adjusting to find θ in the required interval. Always check that solutions lie within the given interval and consider the periodic nature of the functions.

What earns the marks

• Find the principal value using inverse trigonometric functions, and use the symmetry of the sine, cosine and tangent graphs to find all solutions in the given interval.
• Solve quadratic equations in sin θ, cos θ or tan θ by factorising or using the quadratic formula, and reject any roots outside the range −1 ≤ sin θ ≤ 1 or −1 ≤ cos θ ≤ 1.
• For equations involving multiples of the angle, such as sin nθ = k, find all solutions for nθ in an extended interval, then divide by n to obtain θ in the required interval.
• Use exact values for standard angles (0°, 30°, 45°, 60°, 90° and their radian equivalents) where appropriate.
• Check all solutions satisfy the original equation and lie within the specified interval, and present them clearly.

Common mistakes and how to put them right

• Forgetting to find all solutions in the interval, only giving the principal value from the calculator.

  Put it right: To use the symmetry of the graphs to find additional solutions.

• When solving sin θ = k, incorrectly using the negative of the principal value for the second solution.

  Put it right: For sine, the second solution in 0° to 360° is 180° − θ, not −θ.

• For equations like sin 2θ = k, finding solutions for 2θ but forgetting to divide by 2 to get θ, or not adjusting the interval for 2θ correctly.

• Rejecting valid solutions when solving quadratic equations in cos θ, such as discarding a root because it is negative when the interval includes angles where cosine is negative.

Exam tips

• Always sketch the relevant trigonometric graph over the given interval to visualise how many solutions to expect and where they lie.
• When solving equations with multiples of the angle, adjust the interval for the multiple first, solve, then divide by the multiplier to return to the original variable.
• For quadratic equations, clearly show your factorisation or use of the quadratic formula, and state any rejected roots with a reason.
• Give answers in the required units (degrees or radians) and to the required accuracy, and check they lie within the interval.